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WIP: Renakhna (and questions about map overlap)

This is my first (sort of) completed map in my first conworlding project. It's an area called Renakhna - a peninsula in the north of Zaebech, which is in turn one of the continents on Kaliros - my new world. It was created using Fractal Terrains, Adobe Illustrator CS5 and a little bit of Adobe Photoshop CS5.

It needs more work. You'll notice that the southern nations haven't been finished. This is because I decided to work on them in the next map - to the south west of this one - and then copy the details onto this one. I've been very loosely using this awesome tutorial by HandsomeRob but I'm stuck on step 3/4 of his tutorial. My next map overlaps a lot with this one. It looks like this:

The problem comes when I try to copy the detail from the Renakhna map onto this one. There's quite a lot of distortion due to the curvature of the planet because I'm using a Lambert Azimuthal Equal-Area projection. I don't want to have to start again using in different kind of projection, although I realise Mercator would probably work better. Does anyone have any ideas?

I'd also like some feedback on the map if you have something to say and any questions are of course welcome.

I got confused on that tut as well but HandsomeRob hasn't been seen in a long while so we're just guessing about how to go about fixing things. As for projections, I know that someone will be along shortly to answer that part for ya.

If the radiance of a thousand suns was to burst at once into the sky, that would be like the splendor of the Mighty One...I am become Death, the Shatterer of worlds.
-J. Robert Oppenheimer (father of the atom bomb) alluding to The Bhagavad Gita (Chapter 11, Verse 32)

Awesome. I'd really love to find out if there's an easy way to copy all of that detail across and get it in the right proportions. If there's anyone out there who thinks this can't be done though, I'd like to know, because I'll have to find a way around it. Changing to another projection now would be so much effort. Besides, I like the curved look of the LAEA projection.

Thanks. I named the cities using a table of syllables which I created and a random number generator. They didn't just all come off the top of my head, but I've created lots of different languages like this. Renakhnans all speak different dialects of Turso-Kilanean, but when I get round to other maps the languages will mix a bit more. I will probably come back and rename places once I've fleshed out the history too I think some Renakhnan countries have been under military occupation a lot so will probably have a few places with foreign names.

Still stuck on the copying across though. I really hope there's a solution.